Linearly constrained evolutions of critical points and an application to cohesive fractures

TL;DR

This paper introduces a new constructive method for evolving critical points of energy functionals, enabling efficient numerical implementation and proving a discrete-to-continuum limit in cohesive fracture models.

Contribution

It presents a novel approach to define evolutions of critical points that is different from viscosity methods, applicable to nonsmooth, nonconvex energies, with proven convergence in fracture modeling.

Findings

Numerical experiments confirm crack initiation when stress exceeds a material-dependent threshold.

The method provides a consistent discrete-to-continuum limit for cohesive fracture models.

The approach is efficient and applicable to finite-dimensional and certain infinite-dimensional problems.

Abstract

We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite dimensional. Nevertheless, in the infinite dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one and two dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.